There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows:
"Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue."
Surprisingly, this is still open (to my knowledge) for the general case. The only settled cases are the triangles and the quadrilaterals (see Henrot's survey). Is there any progress on the general case?